1. COMPARING MODELS A cup of water at an initial temperature of is placed in a room at a constant…

1. COMPARING MODELS A cup of water at an initial temperature of is placed in a room at a
constant temperature of The temperature of the water is measured every 5 minutes
during a half-hour period. The results are recorded as ordered pairs of the
form where is the time (in minutes) and is the temperature (in degrees
Celsius).

(a) The graph of the model for the
data should be asymptotic with the graph of the temperature of the

room. Subtract the room
temperature from each of the temperatures in the ordered pairs. Use a graphing utility
to plot the data points and

(b) An exponential model

1. COMPARING MODELS A cup of water at an initial temperature of is placed in a room at a
constant temperature of The temperature of the water is measured every 5 minutes
during a half-hour period. The results are recorded as ordered pairs of the
form where is the time (in minutes) and is the temperature (in degrees
Celsius).

(a) The graph of the model for the
data should be asymptotic with the graph of the temperature of the

room. Subtract the room
temperature from each of the temperatures in the ordered pairs. Use a graphing utility
to plot the data points and

(b) An exponential model for the
data is given by Solve for and graph the model. Compare the result with the
plot of the original data.

(c) Take the natural logarithms of
the revised temperatures. Use a graphing utility to plot the points and observe
that the points appear to be linear. Use the regression feature of the
graphing utility to fit a line to these data. This resulting line has the form
Solve for and verify that the result is equivalent to the model in part (b).

(d) Fit a rational model to the
data. Take the reciprocals of the -coordinates of the revised data points to
generate the points Use a graphing utility to graph these points and observe
that they appear to be linear. Use the regression feature of a graphing
utility to fit a line to these data. The resulting line has the form Solve for
and use a graphing utility to graph the rational function and the original data
points.

(e) Why did taking the logarithms
of the temperatures lead to a linear scatter plot? Why did taking the
reciprocals of the temperatures lead to a linear scatter plot?